Final answer:
Torque is calculated using the formula τ = r * F * sin(θ), where r is the distance from the pivot point, F is the force, and θ is the angle relative to the lever arm's axis. For a force applied perpendicularly, the maximum torque is achieved. Coordinate direction angles would require more specific information about the force and radius vector's directions.
Step-by-step explanation:
The concept being discussed in these problems is torque, which is a measure of the rotational effectiveness of a force. Torque (τ) is calculated using the formula τ = r * F * sin(θ), where r is the radius or distance from the pivot point to the point where the force is applied, F is the magnitude of the force, and θ is the angle of the force relative to the lever arm's axis. In the first scenario, if a 14-N horizontal force is applied perpendicularly to the handle of the socket wrench at a distance from point O, the moment or torque about point O is calculated by simply multiplying the force by the distance, as the sine of 90 degrees (which is the angle between a perpendicular force and the wrench handle) is 1.
To find the magnitude of the torque when a force is applied at an angle, we use the same formula but take into account the angle provided (e.g., 40° or 45°). For example, to calculate the torque with a 20-N force applied at a 40° angle to the wrench handle at a radius of 0.25 m, you would apply the torque formula with sin(40°).
The coordinate direction angles such as α, β, and γ often refer to the angles a vector has relative to the x, y, and z axes, respectively. However, to calculate these angles for the torque, we would need to know the direction of the radius vector r and the force vector F in three-dimensional space, which is not provided in the question. The maximum torque would be achieved when the force is applied perpendicularly to the lever arm, meaning θ = 90 degrees because sin(90°) = 1, which gives the maximum value for the sine function.